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The thin membranes and shells in nature are heterogeneous. They are widely used in surgical simulation, biological techniques, and computer animation. The corresponding surface deformable models can implement dynamic simulations of thin membranes and shells in nature, while most surface deformable models are isotropic and cannot represent thin membranes and shells in nature accurately. Therefore, we propose a novel physically-based heterogeneous deformable surface model. By utilizing the same B-spline basis functions or the parameter space of surfaces' geometric representations, we implement material modeling and propose the representations of surfaces with material variations with composite or continuous material functions. Then, we propose a novel physically-based elastic deformable surface model that constructs infinitesimal elements in the parameter space and employs elasticity to analyze their deformation. The corresponding elastic potential energy function is only related to surfaces' continuous representations, and our model avoids the computation error caused by meshes' quality and large rotation of points' frames. We employ isogeometric analysis to solve the dynamic equations derived from our surface model. To demonstrate the validity and reality of our model, several comparison experiments are designed. The corresponding results are in line with expectations and consistent with physical laws.   • We propose representations of heterogeneous objects with component or continuous material functions whose geometric representations are B-spline surfaces by assigning material parameters to control points or subspaces in the parameter space. • We propose a method for deformation analysis of the surface with heterogeneous materials based on infinitesimal elements in the parameter spaces. We employ elasticity to derive the analytical relation between the point's strain and the corresponding infinitesimal element's geometric quantities. • We employ isogeometric analysis to solve the Lagrange dynamic equations that describe heterogeneous objects' motion as the computation methods do not rely on discrete structures. [ABSTRACT FROM AUTHOR]

One prominent step in isogeometric analysis (IGA) is known as domain parameterization, that is, finding a parametric spline representation for a computational domain. Typically, domain parameterization is divided into two separate steps: identifying an appropriate boundary correspondence and then parameterizing the interior region. However, this separation significantly degrades the quality of the parameterization. To attain high-quality parameterization, it is necessary to optimize both the boundary correspondence and the interior mapping simultaneously, referred to as integral parameterization. In a prior research, an integral parameterization approach for planar domains based on neural networks was introduced. One limitation of this approach is that the neural network has no ability of generalization, that is, a network has to be trained to obtain a parameterization for each specific computational domain. In this article, we propose an efficient enhancement over this work, and we train a network which has the capacity of generalization—once the network is trained, a parameterization can be immediately obtained for each specific computational via evaluating the network. The new network greatly speeds up the parameterization process by two orders of magnitudes. We evaluate the performance of the new network on the MPEG data set and a self-design data set, and experimental results demonstrate the superiority of our algorithm compared to state-of-the-art parameterization methods. • A fast parameterization method is proposed via generalization of neural networks. • Our method simultaneously parametrizes the boundary and interior of a domain. • We achieve results as good as the prior work with two orders of magnitude speedup. [ABSTRACT FROM AUTHOR]

The local refinement of PHT-splines (polynomial splines over hierarchical T-meshes) is achieved by a simple cross insertion, which may introduce superfluous control points or coefficients. By allowing split-in-half in mesh refinement, modified hierarchical T-meshes are defined. Using this approach, polynomial splines defined over the modified hierarchical T-meshes (modified PHT-splines) are introduced to increase the flexibility of PHT-splines. Numerical examples demonstrate the advantages of our new splines when applied to surface fitting and isogeometric analysis problems with anisotropic features. • We extend PHT-splines to modified PHT-splines. • We provide a refinement strategy for meshes based on anisotropic features and neighborhood relations. • Numerical experiments show the advantage of modified PHT-splines when dealing with anisotropic problems. [ABSTRACT FROM AUTHOR]

• A NURBS-based finite cell method is proposed for structural topology optimizations. • The framework is able to handle arbitrary shaped trimmed design domains. • A simple and efficient method is proposed to enforce geometric constraints. • Nitsche's method is presented to enforce boundary conditions along trimmed curves. In this work, we present a NURBS-based finite cell approach for topology optimizations of structures with arbitrary shaped trimmed design domains. The combination of isogeometric analysis and finite cell method allows for a topology optimization of trimmed geometries directly derived from CAD systems which eliminates the time consuming preprocessing work such as mesh generations. We further propose a variationally consistent Nitsche's method for the weak enforcement of essential boundary conditions along trimming curves. Independent from the analysis domain, a high order and continuous NURBS represented density field is introduced which not only provides an intrinsic filter but also allows for the realization of multi-resolution topology optimizations. Additionally, we also propose a simple and straight forward method to enforce geometric constraints of arbitrary shapes during topology optimizations. We demonstrate the efficiency and accuracy of the proposed method with a number of benchmark problems and engineering oriented models. [ABSTRACT FROM AUTHOR]

It is well known that star-shaped domains possess particularly simple polar parameterizations, which are formed by the line segments that connect a suitably chosen center with the points on the domain's boundary. The polar parameterization is valid (i.e., regular everywhere except for the center) if the center is located in the kernel of the domain. In the case of a domain with a smooth free-form boundary curve, the kernel is a convex region which is enclosed by the curve and (some of) the boundary's inflection tangents. These parameterizations possess numerous applications, most recently also including domain parameterization in isogeometric analysis. Since the class of star-shaped domains is quite limited, we propose to increase the flexibility of the underlying polar parameterizations by considering circular arcs that connect the center with the points on the domain's boundary. Parameterizations that are regular everywhere except at the center are said to form an arc fibration of a planar domain. We analyze the existence of an arc fibration with a given center and present an algorithm that computes it in the affirmative case. In addition, we explore the arc fibration kernel that contains the suitable center points. [ABSTRACT FROM AUTHOR]

3D objects, modeled using Computer Aided Geometric Design (CAGD) tools, are traditionally represented using a boundary representation (B-rep), and typically use spline functions to parameterize these boundary surfaces. However, recent development in physical analysis, in isogeometric analysis (IGA) in specific, necessitates a volumetric parametrization of the interior of the object. IGA is performed directly by integrating over the spline spaces of the volumetric spline representation of the object. Typically, tensor-product B-spline trivariates are used to parameterize the volumetric domain. A general 3D object, that can be modeled in contemporary B-rep CAD tools, is typically represented using trimmed B-spline surfaces. In order to capture the generality of the contemporary B-rep modeling space, while supporting IGA needs, Massarwi and Elber (2016) proposed the use of trimmed trivariates volumetric elements. However, the use of trimmed geometry makes the integration process more difficult since integration over trimmed B-spline basis functions is a highly challenging task Xu et al. (2017). In this work, we propose an algorithm that precisely decomposes a trimmed B-spline trivariate into a set of (singular only on the boundary) tensor-product B-spline trivariates, that can be utilized to simplify the integration process, in IGA. The trimmed B-spline trivariate is first subdivided into a set of trimmed Bézier trivariates, at all its internal knots. Then, each trimmed Bézier trivariate, is decomposed into a set of mutually exclusive tensor-product B-spline trivariates, that precisely cover the entire trimmed domain. This process, denoted untrimming , can be performed in either the Euclidean space or the parametric space of the trivariate. We present examples of the algorithm on complex trimmed trivariates' based geometry, and we demonstrate the effectiveness of the method by applying IGA over the (untrimmed) results. [ABSTRACT FROM AUTHOR]

Among the bivariate subdivision schemes available, spline-based schemes, such as Catmull-Clark and Loop, are the most commonly used ones. These schemes have known continuity and can be evaluated at arbitrary parameter values. In this work, we develop a C 1 spline-based scheme based on cubic half-box splines. Although the individual surface patches are triangular, the associated control net is three-valent and thus consists in general of mostly hexagons. In addition to introducing stencils that can be applied in extraordinary regions of the mesh, we also consider boundaries. Moreover, we show that the scheme exhibits ineffective eigenvectors. Finally, we briefly consider architectural geometry and isogeometric analysis as selected applications. [ABSTRACT FROM AUTHOR]

• We address the numerical solution for higher order partial differential equations. • Weighted extended THB-splines are constructed for analysis. • A hierarchical extension mechanism is proposed. • The Biharmonic equation is solved numerically on 2D implicit domains. We study the numerical solution for higher order partial differential equations defined on implicit domains. We represent isogeometric analysis on implicit domains and construct weighted extended truncated hierarchical B-splines. Specifically, the spline basis functions are classified, according to the cells occupied by their support inside the domain, using a level by level classification mechanism. A hierarchical extension is then proposed to connect basis functions with small support in the computational domain to more stable basis inside the domain. The basis functions are formulated to acquire local adaptive refinement and preserve stability to release the full approximation power. A numerical illustration is performed to solve the Biharmonic equation on a number of implicit domains with holes and reentrant corners. The error results and condition numbers of the numerical examples reflect the potential of the approach. [ABSTRACT FROM AUTHOR]

Highlights • Investigation of the C 1 -smooth space of trilinearly parameterized volumetric two-patch domains. • Computation of dimension of the space. • Construction of explicitly given and locally supported basis functions of the space. • A first possible extension of the approach to more general two-patch domains than trilinear ones. Abstract We study the C 1 -smooth isogeometric spline space over a specific class of unstructured hexahedral meshes, namely over the class of trilinearly parameterized volumetric two-patch domains. Recently, the structure of this space was experimentally analyzed in Birner et al. (2018) by numerically computing a basis and the dimension of this space. In this work, we develop the theoretical framework to explore the C 1 -smooth isogeometric space. Amongst others, we use the framework to prove the numerically obtained dimension from Birner et al. (2018) and to describe a simple explicit basis construction which consists of locally supported functions. [ABSTRACT FROM AUTHOR]

Abstract Multi-patch spline parametrizations are used in geometric design and isogeometric analysis to represent complex domains. Typically, quadrilateral patches are adopted in both frameworks. We consider the particular class of multi-patch parametrizations that are analysis-suitable G 1 (AS- G 1), which is a specific geometric continuity definition which allows to construct, on the multi-patch domain, C 1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016). It was demonstrated in Kapl et al. (2018) that AS- G 1 multi-patch parametrizations are suitable for modeling complex planar multi-patch domains. We construct a local basis, and an associated dual basis, for a specific C 1 isogeometric spline space A over a given AS- G 1 multi-patch parametrization. The space A is C 1 across interfaces and C 2 at all vertices, and is therefore a subspace of the entire C 1 isogeometric space V 1. At the same time, A allows optimal approximation of traces and normal derivatives along the interfaces and reproduces all derivatives up to second order at the vertices. In contrast to V 1 , the dimension of A does not depend on the domain parametrization. This paper also contains numerical experiments which exhibit the optimal approximation order in L 2 and L ∞ of the isogeometric space A and demonstrate the applicability of our approach for isogeometric analysis. Highlights • We define a C1-smooth isogeometric subspace over multi-patch domains. • The isogeometric subspace is C1 across edges and C2 at vertices. • We present a basis construction for the isogeometric subspace. • The isogeometric subspace displays optimal approximation properties. [ABSTRACT FROM AUTHOR]

Abstract The generation of an analysis-suitable computational grid from a description of no more than its boundaries is a common problem in numerical analysis. Most classical meshing techniques for finite-volume, finite-difference or finite-element applications such as the Advancing Front Method (Schöberl, 1997), Delaunay Triangulation (Triangle, 1996) and elliptic or hyperbolic meshing schemes (Thompson et al., 1998) operate with linear or multi-linear but straight-sided elements for the generation of structured and unstructured meshes, respectively, whereas the generation of high-quality curved meshes is still considered a major challenge. A recent development is the introduction of Isogeometric Analysis (IgA) (Hughes et al., 2005), which can be considered as a natural high-order generalisation of the finite-element method. A description of the geometry Ω ¯ is accomplished via a mapping operator x : Ω ˆ → Ω that maps the unit hypercube in R n onto an approximation Ω of Ω ¯ utilizing a linear combination of higher-order spline functions. The numerical simulation is then carried out in the computational domain Ω ˆ via a ‘ pull back ’ using the mapping operator x. The advantage is that the flexibility of higher-order spline-functions usually allows for an accurate description of Ω ¯ with much fewer elements which can significantly reduce the computational effort required for this step compared to traditional low-order methods. Furthermore, an analytical description of the geometry can be turned back into a traditional (structured or unstructured) grid by performing a large number of function evaluations in x. This can, for instance, be utilized for local refinement without the need for remeshing. A potential drawback of curved instead of linear elements is that the meshing techniques required for the creation of folding-free mappings tend to be more sophisticated and that it is a less trivial task to verify that the resulting mapping is indeed bijective. For the purpose of creating folding-free mappings utilizing spline-functions, we will present an algorithm adopting the principles of Elliptic Grid Generation (EGG) whose basic principles have been adapted to meet the needs of IgA. In R 2 , EGG has particularly appealing properties since bijectivity of the resulting mapping is guaranteed as long as the numerical accuracy to the computational approach of the mapping is sufficient. We will present an algorithm that is capable of generating folding-free mappings from a large number of geometry contours, including complicated geometries from industrial applications with extreme aspect ratios. This is accomplished by combining EGG with automatized reparameterization techniques and a sophisticated numerical approach for solving resulting governing (nonlinear) equations. The algorithm is equipped with the means to verify the bijectivity of the resulting mapping and with automatized defect-correction methods in case a violation of bijectivity is detected. Furthermore, we will present possible strategies for the generation of folding-free mappings for certain types of volumetric geometries by combining EGG with transfinite-interpolation as well as a number of other applications such as time-dependent settings. All applications are provided with example geometries. [ABSTRACT FROM AUTHOR]

Common approximation tools return low-order approximations in the vicinities of singularities. Most prior works solve this problem for univariate functions. In this work we introduce a method for approximating non-smooth multivariate functions of the form f = g + r + where g , r ∈ C M + 1 ( R n ) and the function r + is defined by r + ( y ) = { r ( y ) , r ( y ) ≥ 0 0 , r ( y ) < 0 , ∀ y ∈ R n . Given scattered (or uniform) data points X ⊂ R n , we investigate approximation by quasi-interpolation. We design a correction term, such that the corrected approximation achieves full approximation order on the entire domain. We also show that the correction term is the solution to a Moving Least Squares (MLS) problem, and as such can both be easily computed and is smooth. Last, we prove that the suggested method includes a high-order approximation to the locations of the singularities. [ABSTRACT FROM AUTHOR]

The aim of this paper is a construction of parametric polynomial interpolants of a circular arc possessing maximal geometric smoothness. Two boundary points of a circular arc are interpolated together with higher order geometric data. Construction of interpolants is done via a complex factorization of the implicit unit circle equation. The problem is reduced to solving only one nonlinear equation determined by a monotone function and the existence of the solution is proven for any degree of the interpolating polynomial. Precise starting points for the Newton–Raphson type iteration methods are provided and the best solutions are then given in a closed form. Interpolation by parametric polynomials of degree up to six is discussed in detail and numerical examples confirming theoretical results are included. [ABSTRACT FROM AUTHOR]

A free-form motion can be considered as a smoothly varying rigid-body transformation. Motions can be created by establishing functions in an appropriate space of matrices. While a smooth motion is created, the geometry of the motion itself is not always immediately clear. In a geometric algebra environment, motions can be created using extensions of the ideas of Bézier and B-spline curves and the geometric significance of the construction is clearer. A motion passing through given precision poses can be obtained by direct analogy with the curve approach. This paper considers the more difficult problem of dealing additionally with velocity constraints at the ends of the motion: here the analogy is less obvious. A geometric construction for the end pairs of control poses is established and is demonstrated by creating motions satisfying given pose and velocity constraints. [ABSTRACT FROM AUTHOR]

Construction of spline surfaces from given boundary curves is one of the classical problems in computer aided geometric design, which regains much attention in isogeometric analysis in recent years and is called domain parameterization. However, for most of the state-of-the-art parameterization methods, the rank of the spline parameterization is usually large, which results in higher computational cost in solving numerical PDEs. In this paper, we propose a low-rank representation for the spline parameterization of planar domains using low-rank tensor approximation technique, and apply quasi-conformal map as the framework of the spline parameterization. Under given correspondence of boundary curves, a quasi-conformal map with low rank and low distortion between a unit square and the computational domain can be modeled as a non-linear optimization problem. We propose an efficient algorithm to compute the quasi-conformal map by solving two convex optimization problems alternatively. Experimental results show that our approach outperforms previous approaches in producing bijective and low-rank parametric spline representations of planar domains. [ABSTRACT FROM AUTHOR]

In this paper, we develop a domain-decomposed subspace and multigrid solver to analyze the stress distribution for large-scale finite element meshes with millions of degrees of freedom. Through the domain decomposition technique, the shape editing directly updates the data structure of local finite element matrices. Doing so avoids the expensive factorization step in a direct solver and provides users with a progressive feedback of the stress distribution corresponding to the mesh operations: a fast preview is achieved through the subspace solver, and the multigrid solver refines the preview result if the user needs to examine the stress distribution carefully at certain design stages. Our system constructs the subspace for stress analysis using reduced constrained modes and builds a three-level multigrid solver through the algebraic multigrid method. We remove mid-edge nodes and lump unknowns with the Schur complement method. The updating and solving of the large global stiffness matrix are implemented in parallel after the domain decomposition. Experimental results show that our solver outperforms the parallel Intel MKL solver. Speedups of Image 1 can be achieved for large-scale meshes with reasonable pre-computation costs when setting the stopping criterion of the multigrid solver to be 1e−3 relative error. [ABSTRACT FROM AUTHOR]

We propose an algorithm for computing injective IGA-suitable planar parameterizations with as uniform and orthogonal as possible iso-parametric structure. Central to this approach is a PolySquare-enhanced domain partition procedure that decomposes the input complex planar domain into coarse and square-like quad patches. First, we triangulate the input planar domain that may be multiply connected, and deform it to be a PolySquare-like structure by optimizing a boundary alignment energy. Then, the PolySquare-like structure is pixelated to make the input domain be a quadrilateral mesh that is decimated to generate a coarse patch layout, where each patch is an approximately squared quadrilateral. Finally, the sparse patches are subdivided to represent the input domain and produce the resulting partition. We parameterize each patch with continuous constraints to find the parameterization of the input domain. Compared with existing IGA-suitable planar parameterization methods, our method produces better parameterizations and fewer patches than the ones that also use domain partition strategy. We demonstrate the superiority of our method over various complex domains, including an example containing 30 holes. [ABSTRACT FROM AUTHOR]

Modeling and computing of trivariate parametric volumes is an important research topic in the field of three-dimensional isogeometric analysis. In this paper, we propose two kinds of exact conversion approaches from Bézier tetrahedra to Bézier hexahedra with the same degree by reparametrization technique. In the first method, a Bézier tetrahedron is converted into a degenerate Bézier hexahedron, and in the second approach, a non-degenerate Bézier tetrahedron is converted into four non-degenerate Bézier hexahedra. For the proposed methods, explicit formulas are given to compute the control points of the resulting tensor–product Bézier hexahedra. Furthermore, in the second method, we prove that tetrahedral spline solids with C k -continuity can be converted into a set of tensor–product Bézier volumes with G k -continuity. The proposed methods can be used for the volumetric data exchange problems between different trivariate spline representations in CAD/CAE. Several experimental results are presented to show the effectiveness of the proposed methods. [ABSTRACT FROM AUTHOR]

C 1 continuity is desirable for solving 4th order partial differential equations such as those appearing in Kirchhoff–Love shell models ( Kiendl et al., 2009 ) or Cahn–Hilliard phase field applications ( Gómez et al., 2008 ). Isogeometric analysis provides a useful approach to obtaining approximations with high-smoothness. However, when working with complex geometric domains composed of multiple patches, it is a challenging task to achieve global continuity beyond C 0 . In particular, enforcing C 1 continuity on certain domains can result in “ C 1 -locking” due to the extra constraints applied to the approximation space ( Collin et al., 2016 ). In this contribution, a general framework for coupling surfaces in space is presented as well as an approach to overcome C 1 -locking by local degree elevation along the patch interfaces. This allows the modeling of solutions to 4th order PDEs on complex geometric surfaces, provided that the given patches have G 1 continuity. Numerical studies are conducted for problems involving linear elasticity, Kirchhoff–Love shells and Cahn–Hilliard equation. [ABSTRACT FROM AUTHOR]

This study proposes a new n -sided (n ≥ 4) control point-based surface patch, the blending Bézier patch (BB patch), by constructing corner Bézier surfaces and using Gregory corner blending. A BB patch is defined on a regular polygonal domain with an n -sided control net generalized from quadrilateral Bézier grids, and it reduces to the same degree as a Bézier patch when n = 4. There are two main steps for constructing a BB patch: defining a corner Bézier patch for each corner and blending all corner Bézier patches using rational blending functions. Because the boundary behaviors of the BB patch are similar to those of the Bézier patch, a BB patch can be easily joined to the surrounding Bézier and other BB patches. As an application, we used BB patches to fill the holes with G 2 continuity. • We proposed a new multisided Bézier patch, blending Bézier patch (BB patch), by constructing corner Bézier surfaces and using Gregory corner blending. • BB patch has a simple and intuitive control structure. • BB patch can smoothly connect to adjacent Bézier patches. [ABSTRACT FROM AUTHOR]

In this paper, we present convergence rates for solving elliptic boundary value problems with singular parameterizations in isogeometric analysis. First, the approximation errors with the L 2 ( Ω ) -norm and the H 1 ( Ω ) -seminorm are estimated locally. The impact of singularities is considered in this framework. Second, the convergence rates for solving PDEs with singular parameterizations are discussed. These results are based on a weak solution space that contains all of the weak solutions of elliptic boundary value problems with smooth coefficients. For the smooth weak solutions obtained by isogeometric analysis with singular parameterizations and the finite element method, both are shown to have the optimal convergence rates. For non-smooth weak solutions, the optimal convergence rates are reached by setting proper singularities of a controllable parameterization, even though convergence rates are not optimal by finite element method, and the convergence rates by isogeometric analysis with singular parameterizations are better than the ones by the finite element method. [ABSTRACT FROM AUTHOR]

Computational fluid dynamics (CFD) simulations of flow over complex objects have been performed traditionally using fluid-domain meshes that conform to the shape of the object. However, creating shape conforming meshes for complicated geometries such as automobiles require extensive geometry preprocessing. This process is usually tedious and requires modifying the geometry, including specialized operations such as defeaturing and filling of small gaps. Hsu et al. (2016) developed a novel immersogeometric fluid-flow method that does not require the generation of a boundary-fitted mesh for the fluid domain. However, their method used the NURBS parameterization of the surfaces for generating the surface quadrature points to enforce the boundary conditions, which required the B-rep model to be converted completely to NURBS before analysis can be performed. This conversion usually leads to poorly parameterized NURBS surfaces and can lead to poorly trimmed or missing surface features. In addition, converting simple geometries such as cylinders to NURBS imposes a performance penalty since these geometries have to be dealt with as rational splines. As a result, the geometry has to be inspected again after conversion to ensure analysis compatibility and can increase the computational cost. In this work, we have extended the immersogeometric method to generate surface quadrature points directly using analytic surfaces. We have developed quadrature rules for all four kinds of analytic surfaces: planes, cones, spheres, and tori. We have also developed methods for performing adaptive quadrature on trimmed analytic surfaces. Since analytic surfaces have frequently been used for constructing solid models, this method is also faster to generate quadrature points on real-world geometries than using only NURBS surfaces. To assess the accuracy of the proposed method, we perform simulations of a benchmark problem of flow over a torpedo shape made of analytic surfaces and compare those to immersogeometric simulations of the same model with NURBS surfaces. We also compare the results of our immersogeometric method with those obtained using boundary-fitted CFD of a tessellated torpedo shape, and quantities of interest such as drag coefficient are in good agreement. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of a truck that has a large percentage of analytic surfaces. Using analytic surfaces over NURBS avoids unnecessary surface type conversion and significantly reduces model-preprocessing time, while providing the same accuracy for the aerodynamic quantities of interest. [ABSTRACT FROM AUTHOR]

We study the dimension and construct a basis for C 1 -smooth isogeometric function spaces over two-patch domains. In this context, an isogeometric function is a function defined on a B-spline domain, whose graph surface also has a B-spline representation. We consider constructions along one interface between two patches. We restrict ourselves to a special case of planar B-spline patches of bidegree ( p , p ) with p ≥ 3 , so-called analysis-suitable G 1 geometries, which are derived from a specific geometric continuity condition. This class of two-patch geometries is exactly the one which allows, under certain additional assumptions, C 1 isogeometric spaces with optimal approximation properties (cf. Collin et al., 2016 ). Such spaces are of interest when solving numerically fourth-order PDE problems, such as the biharmonic equation, using the isogeometric method. In particular, we analyze the dimension of the C 1 -smooth isogeometric space and present an explicit representation for a basis of this space. Both the dimension of the space and the basis functions along the common interface depend on the considered two-patch parameterization. Such an explicit, geometry dependent basis construction is important for an efficient implementation of the isogeometric method. The stability of the constructed basis is numerically confirmed for an example configuration. [ABSTRACT FROM AUTHOR]

The space of C 1 cubic Clough–Tocher splines is a classical finite element approximation space over triangulations for solving partial differential equations. However, for such a space there is no B-spline basis available, which is a preferred choice in computer aided geometric design and isogeometric analysis. A B-spline basis is a locally supported basis that forms a convex partition of unity. In this paper, we explore several alternative C 1 cubic spline spaces over triangulations equipped with a B-spline basis. They are defined over a Powell–Sabin refined triangulation and present different types of C 2 super-smoothness. The super-smooth B-splines are obtained through an extraction process, i.e., they are expressed in terms of less smooth basis functions. These alternative spline spaces maintain the same optimal approximation power as Clough–Tocher splines. This is illustrated with a selection of numerical examples in the context of least squares approximation and finite element approximation for second and fourth order boundary value problems. [ABSTRACT FROM AUTHOR]

Constructing an analysis-suitable parameterization for the computational domain from its boundary representation plays a crucial role in the isogeometric design-through-analysis pipeline. PDE-based elliptic grid generation is an effective method for generating high-quality parameterizations with rapid convergence properties for the planar case. However, it may generate non-uniform grid lines, especially near the concave/convex parts of the boundary. In the present work, we introduce a novel scaled discretization of harmonic mappings in the Sobolev space H 1 to remit it. Analytical Jacobian matrices for the involved nonlinear equations are derived to accelerate the computation. To enhance the numerical stability and the speed of convergence, we propose a simple and yet effective preconditioned Anderson acceleration framework instead of using computationally expensive Newton-type iteration. Three preconditioning strategies are suggested, namely diagonal Jacobian, block-diagonal Jacobian, and full Jacobian. Furthermore, we discuss a delayed update strategy of the preconditioner, i.e., the preconditioner is updated every few steps to reduce the computational cost per iteration. Numerical experiments demonstrate the effectiveness and efficiency of our improved parameterization approach and the computational efficiency of our preconditioned Anderson acceleration scheme. [ABSTRACT FROM AUTHOR]

We study the effects of curvature on the energy absorption characteristics of cylindrical corrugated tubes under compression by isogeometric analysis and experiments. The corrugated volume is constructed by revolving a wavelike B-spline profile surface about the vertical axis. The curvature at peaks of the profile curve is gradually increased from a smaller value to a larger value while the wavelength and amplitude are kept unchanged. This leads to three curvature distribution patterns, which are found to significantly affect the energy absorption characteristics of the corrugated tube. Examples show that the tube with the largest curvature pattern absorbs around 21% more energy than that with the smallest curvature pattern. [ABSTRACT FROM AUTHOR]

Based on Grimm and Hughes (1995) we introduce and study a mathematical framework for analysis-suitable unstructured B-spline spaces. In this setting the parameter domain has a manifold structure which allows for the definition of function spaces such as, for instance, B-splines over multi-patch domains with extraordinary points or analysis-suitable unstructured T-splines. Within this framework, we generalize the concept of dual-compatible B-splines (developed for structured T-splines in Beirão da Veiga et al. (2013) ). This allows us to prove the key properties that are needed for isogeometric analysis, such as linear independence and optimal approximation properties for h -refined meshes. [ABSTRACT FROM AUTHOR]

An adaptive isogeometric method based on d -variate hierarchical spline constructions can be derived by considering a refine module that preserves a certain class of admissibility between two consecutive steps of the adaptive loop ( Buffa and Giannelli, 2016 ). In this paper we provide a complexity estimate, i.e., an estimate on how the number of mesh elements grows with respect to the number of elements that are marked for refinement by the adaptive strategy. Our estimate is in the line of the similar ones proved in the context of adaptive finite element methods. [ABSTRACT FROM AUTHOR]

One key feature of isogeometric analysis is that it allows smooth shape functions. Indeed, when isogeometric spaces are constructed from p -degree splines (and extensions, such as NURBS), they enjoy up to C p − 1 continuity within each patch. However, global continuity beyond C 0 on so-called multi-patch geometries poses some significant difficulties. In this work, we consider planar multi-patch domains that have a parametrization which is only C 0 at the patch interface. On such domains we study the h -refinement of C 1 -continuous isogeometric spaces. These spaces in general do not have optimal approximation properties. The reason is that the C 1 -continuity condition easily over-constrains the solution which is, in the worst cases, fully locked to linears at the patch interface. However, recently ( Kapl et al., 2015b ) has given numerical evidence that optimal convergence occurs for bilinear two-patch geometries and cubic (or higher degree) C 1 splines. This is the starting point of our study. We introduce the class of analysis-suitable G 1 geometry parametrizations, which includes piecewise bilinear parametrizations. We then analyze the structure of C 1 isogeometric spaces over analysis-suitable G 1 parametrizations and, by theoretical results and numerical testing, discuss their approximation properties. We also consider examples of geometry parametrizations that are not analysis-suitable, showing that in this case optimal convergence of C 1 isogeometric spaces is prevented. [ABSTRACT FROM AUTHOR]

We present a new method for immersogeometric fluid flow analysis that directly uses the CAD boundary representation (B-rep) of a complex object and immerses it into a locally refined, non-boundary-fitted discretization of the fluid domain. The motivating applications include analyzing the flow over complex geometries, such as moving vehicles, where the detailed geometric features usually require time-consuming, labor-intensive geometry cleanup or mesh manipulation for generating the surrounding boundary-fitted fluid mesh. The proposed method avoids the challenges associated with such procedures. A new method to perform point membership classification of the background mesh quadrature points is also proposed. To faithfully capture the geometry in intersected elements, we implement an adaptive quadrature rule based on the recursive splitting of elements. Dirichlet boundary conditions in intersected elements are enforced weakly in the sense of Nitsche's method. To assess the accuracy of the proposed method, we perform computations of the benchmark problem of flow over a sphere represented using B-rep. Quantities of interest such as drag coefficient are in good agreement with reference values reported in the literature. The results show that the density and distribution of the surface quadrature points are crucial for the weak enforcement of Dirichlet boundary conditions and for obtaining accurate flow solutions. Also, with sufficient levels of surface quadrature element refinement, the quadrature error near the trim curves becomes insignificant. Finally, we demonstrate the effectiveness of our immersogeometric method for high-fidelity industrial scale simulations by performing an aerodynamic analysis of an agricultural tractor directly represented using B-rep. [ABSTRACT FROM AUTHOR]

Building on a result of U. Reif on removable singularities, we construct C 1 bi-3 splines that may include irregular points where less or more than four tensor-product patches meet. The resulting space complements PHT splines, is refinable and the refined spaces are nested, preserving for example surfaces constructed from the splines. As in the regular case, each quadrilateral has four degrees of freedom, each associated with one spline and the splines are linearly independent. Examples of use for surface construction and isogeometric analysis are provided. [ABSTRACT FROM AUTHOR]

Cubic Hermite hexahedral finite element meshes have some well-known advantages over linear tetrahedral finite element meshes in biomechanical and anatomic modeling using isogeometric analysis. These include faster convergence rates as well as the ability to easily model rule-based anatomic features such as cardiac fiber directions. However, it is not possible to create closed complex objects with only regular nodes; these objects require the presence of extraordinary nodes (nodes with 3 or > = 5 adjacent elements in 2D) in the mesh. The presence of extraordinary nodes requires new constraints on the derivatives of adjacent elements to maintain continuity. We have developed a new method that uses an ensemble coordinate frame at the nodes and a local-to-global mapping to maintain continuity. In this paper, we make use of this mapping to create cubic Hermite models of the human ventricles and a four-chamber heart. We also extend the methods to the finite element equations to perform biomechanics simulations using these meshes. The new methods are validated using simple test models and applied to anatomically accurate ventricular meshes with valve annuli to simulate complete cardiac cycle simulations. [ABSTRACT FROM AUTHOR]

In this article we provide the characterization of analysis suitable T-spline spaces ( Beirão da Veiga et al., 2013 ) as the space of piecewise polynomials with appropriate linear constrains on the subdomain interfaces. We describe AST-meshes for which the linear constrains are equivalent to smoothness conditions and provide examples showing that this is not always the case. [ABSTRACT FROM AUTHOR]

This paper proposes a methodology to create a hybrid volumetric representation from a 2-manifold without boundaries represented with untrimmed B-spline surfaces. The product consists of trivariate tensor product B-splines near the boundary and unstructured higher-order Bézier tetrahedral elements in the interior of the object with C 0 smoothness across their interfaces. The B-spline elements are constructed by offsetting the input surface into its interior. Then, an intermediate interface consisting of Bézier triangles is built to match the inner boundary of the B-spline representation. The rest of the space, bounded by the intermediate interface, is then filled with unstructured Bézier tetrahedral elements. Our approach to constructing stiffness and mass matrices takes into account the dependencies of interior Bézier tetrahedral elements on exterior B-spline elements along their interface, so that C 0 smoothness is automatically maintained when performing computer graphics simulations, representing material properties, or performing isogeometric analysis. We apply our methodology on a variety of 3-D objects and demonstrate a fast convergence rate with our 2-D studies. [ABSTRACT FROM AUTHOR]

With the advent of isogeometric analysis, the modeling of spline solids became an important topic. In this paper, we present a discrete volume parameterization method for tetrahedral (tet) mesh models and an iterative fitting algorithm with a B-spline solid. The discrete volume parameterization method maps the vertices of a tet mesh into a parameter domain by solving a system of linear equations. Each equation is explicitly constructed for an inner vertex in terms of the geometric information adjacent to the inner vertex. Moreover, we show the validity of the parameterization system of linear equations thus constructed. Next, because the number of tet mesh vertices is usually very large, we develop an iterative algorithm for fitting a tet mesh with a B-spline solid. The iterative algorithm exploits the geometric information of the control hexahedral (hex) mesh and the local support property of the spline function, so the total amount of computation in each iteration is unchanged when the number of control hex mesh vertices of the B-spline solid is increased. Therefore, the iterative fitting algorithm performs very well in incremental fitting of a tet mesh with a large number of vertices. Finally, four experimental examples presented in this paper show the efficiency and effectiveness of the developed algorithms. [ABSTRACT FROM AUTHOR]

Nitsche method application in non-conforming plate is presented in the context of isogeometric analysis. Reissner–Mindlin plate theory is employed to build governing equation and stiffness matrix. We use this theory to solve the elasticity problems of various classical plate models, and compare the obtained results to those from single-patch models and the exact solutions in Kirchhoff theory. The solutions of problem involving the use of complex model are as well obtained using the same Reissner–Mindlin theory and compared to the results from finite element method. All models are built with NURBS (non-uniform rational B-spline) patches with non-conforming mesh along the common boundaries. The algorithms of knot insertion and order elevation are applied to enrich the basis functions of NURBS patches. The results of numerical examples show the accuracy, robustness and high convergence rate of this method. [ABSTRACT FROM AUTHOR]

Isogeometric analysis uses spline parameterizations to describe the geometry of the physical domain. If such a parameterization is not available, it has to be generated from the domain boundaries. The construction of a good parameterization is crucial since it strongly influences the accuracy of the subsequent analysis. It is of interest to use adaptive techniques such as hierarchical splines for the parameterization, since this facilitates an accurate representation of detailed geometries and potentially improves the efficiency of the subsequent numerical simulation. We use truncated hierarchical (TH) B-splines to generate hierarchical spline spaces, since these functions possess many useful properties, such as linear independence and the fact that they form a non-negative partition of unity. In order to address the trade-off between computational effort and level of difficulty of a specific instance of the problem, we propose three levels of domain parameterization techniques that are based on THB-splines. [ABSTRACT FROM AUTHOR]

We present an efficient adaptive refinement procedure that preserves analysis-suitability of the T-mesh, that is, the linear independence of the T-spline blending functions. We prove analysis-suitability of the overlays and boundedness of their cardinalities, nestedness of the generated T-spline spaces, and linear computational complexity of the refinement procedure in terms of the number of marked and generated mesh elements. [ABSTRACT FROM AUTHOR]

G k (geometrically continuous surface) constructions were developed to create surfaces that are smooth also at irregular points where, in a quad-mesh, three or more than four elements come together. Isogeometric elements were developed to unify the representation of geometry and of engineering analysis. We show how matched G k constructions for geometry and analysis automatically yield C k isogeometric elements. This provides a formal framework for the existing and any future isogeometric elements based on geometric continuity. [ABSTRACT FROM AUTHOR]

Constructing a high-quality parameterization of a computational domain is a fundamental research problem in isogeometric analysis, which has been extensively investigated so far. However, most of the current approaches employ non-uniform rational B-splines (NURBS) as the geometric representation of the physical domain. NURBS introduce redundant degrees of freedom due to their tensor-product structure. In this paper, we propose a new parameterization method for planar domains by adopting hierarchical B-splines (HB-splines) as the geometric representation that possess local refinement abilities. Starting from an initial parameterization such as a harmonic map, our method repeats the following two steps until a bijective parameterization with low distortion is achieved. First, a non-linear optimization model is proposed to compute a quasi-conformal map represented by HB-splines, and an efficient algorithm is provided to deal with this model by alternatively solving two quadratic optimization problems. Second, the parameterization is refined locally through HB-splines based on the bijectivity and conformal distortion of the parameterization. Several examples are demonstrated to verity the effectiveness and advantages of the proposed approach. • A novel parameterization method for planar domains is presented. • The parameterization is represented by HB-splines and is based on QC maps. • An alternative optimization algorithm is developed to compute the parameterization. • Numerical examples indicate that the new approach is superior to other methods. [ABSTRACT FROM AUTHOR]

We consider isogeometric functions and their derivatives. Given a geometry mapping, which is defined by an n -dimensional NURBS patch in R d , an isogeometric function is obtained by composing the inverse of the geometry mapping with a NURBS function in the parameter domain. Hence an isogeometric function can be represented by a NURBS parametrization of its graph. We take advantage of the projective representation of the NURBS patch as a B-spline patch in homogeneous coordinates. We derive a closed form representation of the graph of a partial derivative of an isogeometric function. The derivative can be interpreted as an isogeometric function of higher degree and lower smoothness on the same piecewise rational geometry mapping, hence the space of isogeometric functions is closed under differentiation. We distinguish the two cases n = d and n < d , with a focus on n = d − 1 in the latter one. As a first application of the presented formula we derive conditions which guarantee C 1 and C 2 smoothness for isogeometric functions on several singularly parametrized planar and volumetric domains as well as on embedded surfaces. It is interesting to note that the presented conditions depend not only on the general structure of the patch, but on the exact representation of the interior of the given geometry mapping. [ABSTRACT FROM AUTHOR]

Abstract: In this work, an extension has been performed on the analysis basis of spline-based meshfree method (SBMFM) to stabilize its solution. The potential weakness of the SBMFM is its numerical instability from using regular grid background mesh. That is, if an extremely small trimmed element is produced by the trimming curves that represent boundaries of the analysis domain, it can induce an excessively large condition number in global system matrix. To resolve the instability problem, the extension technique of the weighted extended B-spline (WEB-spline) is implemented in the SBMFM. The basis functions with very small trimmed supports are extrapolated by neighboring basis functions with some special scheme so that those basis functions can be condensed in the solution process. In order to impose essential boundary conditions in the SBMFM with extended basis, Nitsche's method is implemented. Using numerical examples, the presented SBMFM with extended basis is shown to be valid and effective. Moreover, the condition number of the system is well-managed guaranteeing the stability of the numerical analysis. [Copyright &y& Elsevier]

In this paper, a deep learning framework combined with isogeometric analysis (IGA for short) called IGA-Reuse-Net is proposed for efficient reuse of numerical simulation on a set of topology-consistent models. Compared with previous data-driven numerical simulation methods only for simple computational domains, our method can predict high-accuracy PDE solutions over topology-consistent geometries with complex boundaries. UNet3+ architecture with interlaced sparse self-attention (ISSA) module is used to enhance the performance of the network. In addition, we propose a new loss function that combines a coefficients loss and a numerical solution loss. Several training datasets with topology-consistent models are constructed for the proposed framework. To verify the effectiveness of our approach, two different types of Poisson equations with different source functions are solved on three datasets with different topologies. Our framework can achieve a good trade-off between accuracy and efficiency. It outperforms the physics-informed neural network (PINN for short) model and yields promising results of prediction. • A deep learning framework is proposed for efficient reuse of IGA simulation on topology-consistent parameterizations. • UNet3+ architecture with ISSA module is proposed to enhance the performance of the network. • Several training datasets with topology-consistent models are also constructed for the proposed framework. [ABSTRACT FROM AUTHOR]

Abstract: We address progressive local refinement of splines defined on axes parallel box-partitions and corresponding box-meshes in any space dimension. The refinement is specified by a sequence of mesh-rectangles (axes parallel hyperrectangles) in the mesh defining the spline spaces. In the 2-variate case a mesh-rectangle is a knotline segment. When starting from a tensor-mesh this refinement process builds what we denote an LR-mesh, a special instance of a box-mesh. On the LR-mesh we obtain a collection of hierarchically scaled B-splines, denoted LR B-splines, that forms a nonnegative partition of unity and spans the complete piecewise polynomial space on the mesh when the mesh construction follows certain simple rules. The dimensionality of the spline space can be determined using some recent dimension formulas. [Copyright &y& Elsevier]